Geometry of CR-submanifolds by Aurel Bejancu

By Aurel Bejancu

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9) for any Y E r(D) and X E r(TM). Next, we denote by ri(TM) the real vector space of all tensor fields of type (1, 1) on M. Also, we consider the vector subspace ri(D) of ri(TM) defined by ri(D) = {H E rl(TM); HQY = QHPY = 0, dY E r(TM)}. 11) TH = 2(H - oHo4). 13) where I is the identity morphism on ri(D). 9) we see that SX E r1 (D). 14) We have further that IV ('- 4 4o(VX0)eP) - 0, 0 since VXP - 0. 14) are given by SX = 2 (OX4))o4) + K. 15 where KX is an arbitrary element of )'(D). 16) where K is a tensor field of type (1, 2) on M satisfying K(X, QY) = QK(X, PY) = 0.

30 CHAPCERH §3. 4)-Connections on a CR-Submanifold and CR-Products of Almost Hermitian Manifolds Let M be a CR-submanifold of an almost Hermitian manifold N. 1 is an f-structure. A linear connection V on M is called a 4)-connection if 4) is covariantly constant with respect to this connection, that is, we have V Y = 0, X for each X E r(TM). 1 (Bejancu (101 ). 2) 0 where V is a linear connection with respect to which both distributions D and are parallel, P and Q are the projection morphisms to D and respectively and K and S are arbitrary tensor fields of type (1, 2) on M.

The sectional curvature for a quaternion plane is called a quaternion sectional curvature. If the quaternion sectional curvature is a constant c for all quaternion planes and for all points x of N we say that N is a quaternion space form and denote it by N(c). The curvature tensor of a quaternion space form N(c) is given by (X, Y)Z = 4 {g (Y' Z)X - g(X, Z)Y + 3 + E {g(JaY, Z)JaX - g(JaX, Z)JaY + a=1 + 2g(X, JaY)JaZ)), for any X, Y, z E r(TN). 7) Chapter II CR-SUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS §1.

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